We consider now the classical 2D Lid-driven cavity test case. The computational domain is the box \(\Omega = \left(0,1\right)\times \left(0,1\right)\)  which is initialized with a density \(\rho = 1\)  and a velocity \(\mathbf {u} = \mathbf {0}\) . The flow is driven by the upper boundary, whose velocity is set to \(\mathbf {u} = \left(1,0\right)^{T}\) , while on the other three boundaries a no-slip condition is imposed. We set \(Re = 100\)  and \(Ma^2 = 10^{-5}\) . The advantage of the proposed scheme is that the allowed time step is more than 100 times larger compared to that of a fully explicit scheme. Indeed, the time-step chosen is such that the maximum advective Courant number \(C_u\)  is around \(0.12\) , while the maximum Courant number \(C\)  is around 49. The streamlines are shown in Figure REF  and highlight the formation of the main recirculation pattern. A comparison of the horizontal component of the velocity along the vertical middle line and of the vertical component of the velocity along the horizontal middle line with the reference solutions in [1]}, [2]} is also presented.
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